Then I stopped & thought; maybe I shouldn"t be doing that? I am doubting myself! Can anyone confirm if I am doing the right thing lớn solve the equality?
I like to vày tables khổng lồ determine signs. It follows from Bolzano"s Theorem the sign will be kept between the roots. You already factored this và obtained $x=2,x=-2,x=0$.
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$$eginarrayc và (-infty,-2) và (-2,0)& (0,2)&(2,+infty)\ hlinex-2 và - &- & -&+\ hlinex+2 &- &+ &+ &+\ hlinex và - và - &+ &+\ hlinep(x)& -&+&-&+\ hlineendarray$$
Explanation: The vị trí cao nhất row is the real line divided into the intervals by the roots we have. The first three rows of $+/-$ tài khoản for the sign of the factors in each interval, which we obtain by inspection. Finally, the sign of $p$ is obtained by "multiplying" each of the values obtained, so $- imes- imes-=-$, $- imes+ imes +=+$,$&c$.
Try and do the same for $p(x)=x^3 + 4x^2 + 3x - 2$.
A good way to vị this is to lớn draw a number line và mark the places where the polynomial is zero. The sign will be constant in the intervals so created. Check inside each one to see the sign (+ or -). Then it"s easy khổng lồ read the solution.
The polynomial $x^3-4x$ is $0$ at $x=-2, 0,$ và $2.$ At $-3$ the value of the polynomial is $0.$ At $-1,$ the value is $>0.$ At $1$ the value is $ 0$ for$$-22.$$
This is a good approach. You are almost there. A product is positive if it has an even number of negative terms. So it is positive if there are no negative terms, which is if $x gt 2$, or if there are two negative terms ...
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